The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given: \[3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.\]

Find the angle between two common sections of the page $2x+y-z=0$ and the cone $4x^2-y^2+3z^2=0.$

Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$, \[f(m - n + f(n)) = f(m) + f(n).\]

Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x)= (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$. [i]Proposed by Alex Gu[/i]

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

Find three different polynomials $P(x)$ with real coefficients such that $P\left(x^2 + 1\right) = P(x)^2 + 1$ for all real $x$.

Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.

Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n = \frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]R4.16 / P1.4[/b] Adam and Becky are building a house. Becky works twice as fast as Adam does, and they both work at constant speeds for the same amount of time each day. They plan to finish building in $6$ days. However, after $2$ days, their friend Charlie also helps with building the house. Because of this, they finish building in just $5$ days. What fraction of the house did Adam build? [b]R4.17[/b] A bag with $10$ items contains both pencils and pens. Kanye randomly chooses two items from the bag, with replacement. Suppose the probability that he chooses $1$ pen and $1$ pencil is $\frac{21}{50}$ . What are all possible values for the number of pens in the bag? [b]R4.18 / P2.8[/b] In cyclic quadrilateral $ABCD$, $\angle ABD = 40^o$, and $\angle DAC = 40^o$. Compute the measure of $\angle ADC$ in degrees. (In cyclic quadrilaterals, opposite angles sum up to $180^o$.) [b]R4.19 / P2.6[/b] There is a strange random number generator which always returns a positive integer between $1$ and $7500$, inclusive. Half of the time, it returns a uniformly random positive integer multiple of $25$, and the other half of the time, it returns a uniformly random positive integer that isn’t a multiple of $25$. What is the probability that a number returned from the generator is a multiple of $30$? [b]R4.20 / P2.7[/b] Julia is shopping for clothes. She finds $T$ different tops and $S$ different skirts that she likes, where $T \ge S > 0$. Julia can either get one top and one skirt, just one top, or just one skirt. If there are $50$ ways in which she can make her choice, what is $T - S$? [u]Set 5[/u] [b]R5.21[/b] A $5 \times 5 \times 5$ cube’s surface is completely painted blue. The cube is then completely split into $ 1 \times 1 \times 1$ cubes. What is the average number of blue faces on each $ 1 \times 1 \times 1$ cube? [b]R5.22 / P2.10[/b] Find the number of values of $n$ such that a regular $n$-gon has interior angles with integer degree measures. [b]R5.23[/b] $4$ positive integers form an geometric sequence. The sum of the $4$ numbers is $255$, and the average of the second and the fourth number is $102$. What is the smallest number in the sequence? [b]R5.24[/b] Let $S$ be the set of all positive integers which have three digits when written in base $2016$ and two digits when written in base $2017$. Find the size of $S$. [b]R5.25 / P3.12[/b] In square $ABCD$ with side length $13$, point $E$ lies on segment $CD$. Segment $AE$ divides $ABCD$ into triangle $ADE$ and quadrilateral $ABCE$. If the ratio of the area of $ADE$ to the area of $ABCE$ is $4 : 11$, what is the ratio of the perimeter of $ADE$ to the perimeter of $ABCE$? [u]Set 6[/u] [b]R6.26 / P6.25[/b] Submit a decimal n to the nearest thousandth between $0$ and $200$. Your score will be $\min (12, S)$, where $S$ is the non-negative difference between $n$ and the largest number less than or equal to $n$ chosen by another team (if you choose the smallest number, $S = n$). For example, 1.414 is an acceptable answer, while $\sqrt2$ and $1.4142$ are not. [b]R6.27 / P6.27[/b] Guang is going hard on his YNA project. From $1:00$ AM Saturday to $1:00$ AM Sunday, the probability that he is not finished with his project $x$ hours after $1:00$ AM on Saturday is $\frac{1}{x+1}$ . If Guang does not finish by 1:00 AM on Sunday, he will stop procrastinating and finish the project immediately. Find the expected number of minutes $A$ it will take for him to finish his project. An estimate of $E$ will earn $12 \cdot 2^{-|E-A|/60}$ points. [b]R6.28 / P6.28[/b] All the diagonals of a regular $100$-gon (a regular polygon with $100$ sides) are drawn. Let $A$ be the number of distinct intersection points between all the diagonals. Find $A$. An estimate of $E$ will earn $12 \cdot \left(16 \log_{10}\left(\max \left(\frac{E}{A},\frac{A}{E}\right)\right)+ 1\right)^{-\frac12}$ or $0$ points if this expression is undefined. [b]R6.29 / P6.29 [/b]Find the smallest positive integer $A$ such that the following is true: if every integer $1, 2, ..., A$ is colored either red or blue, then no matter how they are colored, there are always 6 integers among them forming an increasing arithmetic progression that are all colored the same color. An estimate of $E$ will earn $12 min \left(\frac{E}{A},\frac{A}{E}\right)$ points or $0$ points if this expression is undefined. [b]R6.30 / P6.30[/b] For all integers $n \ge 2$, let $f(n)$ denote the smallest prime factor of $n$. Find $A =\sum^{10^6}_{n=2}f(n)$. In other words, take the smallest prime factor of every integer from $2$ to $10^6$ and sum them all up to get $A$. You may find the following values helpful: there are $78498$ primes below $10^6$, $9592$ primes below $10^5$, $1229$ primes below $10^4$, and $168$ primes below $10^3$. An estimate of $E$ will earn $\max \left(0, 12-4 \log_{10}(max \left(\frac{E}{A},\frac{A}{E}\right)\right)$ or $0$ points if this expression is undefined. PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that all square matrices of the form (with $a, b \in R$), $$\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$$ form a commutative field $K$ when considering the operations of addition and matrix product. Prove also that if $A \in K$ is an element of said field, there exist two matrices of $K$ such that the square of each is equal to $A$.

Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$

We shall call the triplet of numbers $a, b, c$ of the interval $[-1,1]$ [i]qualitative [/i] if these numbers satisfy the inequality $1 + 2abc\ge a^2 + b^2 + c^2$. Prove that when the triples $a, b, c$, and $x, y, z$ are qualitative, then $ax, by, cz$ is also qualitative.

Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

(a) For given complex numbers $a_1,a_2,a_3,a_4$, we define a function $P:\mathbb C\to\mathbb C$ by $P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z$. Let $w_k=e^{2ki\pi/5}$, where $k=0,\ldots,4$. Prove that $$P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.$$(b) Let $A_1,A_2,A_3,A_4,A_5$ be five points in the plane. A pentagon is inscribed in the circle with center $A_1$ and radius $R$. Prove that there is a vertex $S$ of the pentagon for which $$SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.$$

If $p$ and $q$ are odd integers, prove that the equation $$x^2 + 2px + 2q = 0$$ has no rational roots.

Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .

Given $n^2$ numbers $x_{i,j}$ ($i,j=1,2,...,n$) satisfying the system of $n^3$ equations $$x_{i,j}+x_{j,k}+x_{k,i}=0 \,\,\, (i,j,k = 1,...,n)$$Prove that there exist such numbers $a_1,a_2,...,a_n$, that $x_{i,j}=a_i-a_j$ for all $i,j=1,...n$.

Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Could it happen that from segments of lengths $$\sqrt{a^2 + \frac{2}{3} bc},\quad \sqrt{b^2 + \frac{2}{3} ca}\quad \text{and} \quad \sqrt{c^2 + \frac{2}{3} ab},$$ a right-angled triangle can be formed?

Let $a_1, a_2, \ldots, a_{2016}$ be real numbers such that $9a_i\ge 11a^2_{i+1}$ $(i=,2,\cdots,2015)$. Find the maximum value of $(a_1-a^2_2)(a_2-a^2_3)\cdots (a_{2015}-a^2_{2016})(a_{2016}-a^2_{1}).$

Let $a, b, c$ be real numbers in the interval $[0, 1]$, satisfying $ab + c \le 1$. Find the maximal value of their sum $a + b + c$.

Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. [See example 25 for the meaning of $|x|$.] The number of polynomials with $h=3$ is: $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $